mm-639
===
Subject: Regular Sets
I am totally lost on this one...can someone please help...
Determine whether 1011 belongs to each of these regular sets:
10*1*
0*(10 U 11)*
1(01)*1*
1*01(0 U 1)
(10)*(11)*
1(00)*(11)*
(10)*1011
(1 U 00)(01 U 0)1*
===
Subject: Re: Regular Sets
alt.math.undergrad:
> I am totally lost on this one...can someone please help...
> Determine whether 1011 belongs to each of these regular
sets:
In each case you have to decide whether the string 1011 can
be made to match the indicated pattern. To do this you have
to be able to interpret the notation for regular sets. I'll
do the [CapitalThorn]rst three in some detail and let you try your hand
again on the rest.
> 10*1*
What has to be true of a string in order for it to match
this pattern? It must:
(a) begin with a 1;
(b) then have zero or more 0's; and
(c) [CapitalThorn]nish with zero or more 1's.
Since you can break up 1011 as
1 0 11
(a) (b) (c),
it matches this pattern and does belong to the set.
> 0*(10 U 11)*
This one's a little more complicated. It describes strings
that can be broken into two parts, say uv, where u is in 0*
and v is in (10 U 11)*. Can you split 1011 in such a way?
0* includes only the empty string, 0, 00, 000, 0000, and so
on; since 1011 begins with a 1, the only possible u is the
empty string, so we'll try it. That means that the v will
have to be 1011 itself; is that in (10 U 11)*? To answer
that, you have to understand what's in (10 U 11)*. The
empty string is, but what else there? 10 and 11 are. So
are 1010, 1011, 1110, and 1111. And in general so is any
string that you can build by repeatedly choosing one of 10
and 11 and appending it to one that you already have. The
ones of length 6, for instance, are 101010, 101011, 101110,
101111, 111010, 111011, 111110, and 111111; each of these is
one of the strings of length 4 extended by either 10 or 11
on the right.
In particular, 1011 is in (10 U 11)*, so it's also in
0*(10 U 11)*, with the invisible empty string at its front
matching 0*:
0* (10 U 11)*
- 10 11
> 1(01)*1*
The only way to match a pattern 1 is with the string 1, so
if 1011 is going to match this pattern, it has to be because
the [CapitalThorn]rst 1 matches 1 and the remaining 011 matches (01)*1*.
This means that 011 must decompose into two substrings, say
011 = uv, such that u matches (01)* and v matches 1*. This
is easy: take u = 01 and v = 1. (Remember, (01)* consists
of the empty string, 01, 0101, 010101, and so on: strings of
zero or more copies of 01. Similarly, 1* consists of
strings of zero or more 1's.) Thus, 1011 matches 1(01)*1*
as set out below:
1 (10)* 1*
1 10 11
One hint for the remaining [CapitalThorn]ve: 1011 is in all but one of
the sets.
> 1*01(0 U 1)
> (10)*(11)*
> 1(00)*(11)*
> (10)*1011
> (1 U 00)(01 U 0)1*
Brian
===
Subject: Re: Uniform Convergence
alt.math.undergrad:
[...]
> I am trying to develop logarithms in a different way
> from the classical way of considering the integral of
> 1/x. So I differentiate c^x with respect to x, and get
> the limit as h tends to 0 of f(c,h) multiplied by c^x.
> Assuming you seek rigor here, it seems to me you have
> some problems. What is the de[CapitalThorn]nition of c^x? Why is it
> differentiable?
[...]
> De[CapitalThorn]ne c^x in the standard way as follows:
> Suppose x is a natural number, then c^x is de[CapitalThorn]ned
> inductively as c^1 = c and c^(x+1)=c*c^x (the proof that
> this is well de[CapitalThorn]ned follows from the principle of
> induction trivially).
> Suppose x is an integer. If x is 0, c^x is 1. If x is
> negative, c^x is 1/c^(-x).
> Suppose x is rational. Then x = p/q, p and q integers and
> q non-zero. Then c^x is the solution y of the equation
> y^q = c^p.
To make this rigorous you have to show that if p/q = r/s,
then y^q = c^p and y^s = c^r have the same solutions. You
also have to deal with the fact that y^2 = c^1 has two
solutions (for c > 0, which is presumably the only case that
you're considering). You also have to check that 
it's
consistent with the earlier parts of the de[CapitalThorn]nition, though
that's pretty trivial.
> Suppose x is real. Then there is a sequence of rationals
> x_n tending to x. So c^x is the limit as n tends to
> in[CapitalThorn]nite of c^(x_n).
To make this rigorous you have to prove that the limit
exists and is independent of the particular sequence <x_n>
chosen. You also have to show that this is consistent with
the earlier parts of the de[CapitalThorn]nition.
> Suppose x is complex. Then c^x is exp(x*ln(c)).
How are you de[CapitalThorn]ning the logarithm on C? I thought that
this was in aid of developing the logarithm function. Or
are you just saying that once you have real logs, you'll
develop complex logs in the usual way?
[...]
Brian
===
Subject: Re: Uniform Convergence
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EG4AI14235;
>alt.math.undergrad:
>[...]
> I am trying to develop logarithms in a different way
> from the classical way of considering the integral of
> 1/x. So I differentiate c^x with respect to x, and get
> the limit as h tends to 0 of f(c,h) multiplied by c^x.
> Assuming you seek rigor here, it seems to me you have
> some problems. What is the de[CapitalThorn]nition of c^x? Why is it
> differentiable?
>[...]
> De[CapitalThorn]ne c^x in the standard way as follows:
> Suppose x is a natural number, then c^x is de[CapitalThorn]ned
> inductively as c^1 = c and c^(x+1)=c*c^x (the proof that
> this is well de[CapitalThorn]ned follows from the principle of
> induction trivially).
> Suppose x is an integer. If x is 0, c^x is 1. If x is
> negative, c^x is 1/c^(-x).
> Suppose x is rational. Then x = p/q, p and q integers and
> q non-zero. Then c^x is the solution y of the equation
> y^q = c^p.
>To make this rigorous you have to show that if p/q = r/s,
>then y^q = c^p and y^s = c^r have the same solutions. You
>also have to deal with the fact that y^2 = c^1 has two
>solutions (for c > 0, which is presumably the only case that
>you're considering). You also have to check that 
it's
>consistent with the earlier parts of the de[CapitalThorn]nition, though
>that's pretty trivial.
> Suppose x is real. Then there is a sequence of rationals
> x_n tending to x. So c^x is the limit as n tends to
> in[CapitalThorn]nite of c^(x_n).
>To make this rigorous you have to prove that the limit
>exists and is independent of the particular sequence <x_nchosen. You also have to show that this is consistent with
>the earlier parts of the de[CapitalThorn]nition.
> Suppose x is complex. Then c^x is exp(x*ln(c)).
>How are you de[CapitalThorn]ning the logarithm on C? I thought that
>this was in aid of developing the logarithm function. Or
>are you just saying that once you have real logs, you'll
>develop complex logs in the usual way?
>[...]
>Brian
Oh dear, I hadn't really thought of that last point. But
yeah, I'll do
that - I'll develop ln for real argument and then 
de[CapitalThorn]ne c^x
where c
is real and x complex.
In terms of your earlier points - I had thought of most of
these, but
what do you mean for real numbers where you say you need to
prove that
the limit exists? I am de[CapitalThorn]ning x_n as a rational sequence
converging
to x, so obviously the limit exists, otherwise the sequence
does not
satisfy the de[CapitalThorn]nition!
===
Subject: Re: Uniform Convergence
alt.math.undergrad:
>alt.math.undergrad:
>[...]
> I am trying to develop logarithms in a different way
> from the classical way of considering the integral of
> 1/x. So I differentiate c^x with respect to x, and get
> the limit as h tends to 0 of f(c,h) multiplied by c^x.
> Assuming you seek rigor here, it seems to me you have
> some problems. What is the de[CapitalThorn]nition of c^x? Why is it
> differentiable?
>[...]
> De[CapitalThorn]ne c^x in the standard way as follows:
> Suppose x is a natural number, then c^x is de[CapitalThorn]ned
> inductively as c^1 = c and c^(x+1)=c*c^x (the proof that
> this is well de[CapitalThorn]ned follows from the principle of
> induction trivially).
> Suppose x is an integer. If x is 0, c^x is 1. If x is
> negative, c^x is 1/c^(-x).
> Suppose x is rational. Then x = p/q, p and q integers and
> q non-zero. Then c^x is the solution y of the equation
> y^q = c^p.
>To make this rigorous you have to show that if p/q = r/s,
>then y^q = c^p and y^s = c^r have the same solutions. You
>also have to deal with the fact that y^2 = c^1 has two
>solutions (for c > 0, which is presumably the only case that
>you're considering). You also have to check that 
it's
>consistent with the earlier parts of the de[CapitalThorn]nition, though
>that's pretty trivial.
> Suppose x is real. Then there is a sequence of rationals
> x_n tending to x. So c^x is the limit as n tends to
> in[CapitalThorn]nite of c^(x_n).
>To make this rigorous you have to prove that the limit
>exists and is independent of the particular sequence <x_nchosen. You also have to show that this is consistent with
>the earlier parts of the de[CapitalThorn]nition.
[...]
> In terms of your earlier points - I had thought of most of
these, but
> what do you mean for real numbers where you say you need to
prove that
> the limit exists?
You have to prove that if <x_n : n in N> converges to x,
then <c^{x_n} : n in N> actually converges. This is not
trivial.
[...]
Brian
===
Subject: Decomposition of {0,1} strings problem.
I can't [CapitalThorn]gure this one out.
I need to write a decomposition for all {0,1} strings that
uniquely
creates all strings such that the substring 0110 does not
appear.
Now, I was able to get 011, but I'm unable to 
[CapitalThorn]gure out a
decomposition
for 0110. I came up with a bunch, but none of they are
uniquely created.
Can anybody offer any hints as to how I could do this?
===
Subject: Re: Decomposition of {0,1} strings problem.
alt.math.undergrad:
> I can't [CapitalThorn]gure this one out.
> I need to write a decomposition for all {0,1} strings that
uniquely
> creates all strings such that the substring 0110 does not
appear.
I might be able to help if I were familiar with the
terminology that you're using: what's your 
de[CapitalThorn]nition of a
decomposition?
[...]
Brian
===
Subject: Re: Decomposition of {0,1} strings problem.
For example, so far I have the following:
{1}*({0}({E,1}U{111}{1}*))* and that seems very close, but it
doesn't
account for 011. I can't just add {1}* at the end since it
would no
longer be uniquely created.
> alt.math.undergrad:
>I can't [CapitalThorn]gure this one out.
>I need to write a decomposition for all {0,1} strings that
uniquely
>creates all strings such that the substring 0110 does not
appear.
> I might be able to help if I were familiar with the
> terminology that you're using: what's your 
de[CapitalThorn]nition of a
> decomposition?
> [...]
> Brian
===
Subject: Re: Decomposition of {0,1} strings problem.
Nevermind, I [CapitalThorn]gured it out
You need to add {E,011} to the end
> For example, so far I have the following:
> {1}*({0}({E,1}U{111}{1}*))* and that seems very close, but
it doesn't
> account for 011. I can't just add {1}* at the end since it
would no
> longer be uniquely created.
> alt.math.undergrad:
> I can't [CapitalThorn]gure this one out.
> I need to write a decomposition for all {0,1} strings
that uniquely
> creates all strings such that the substring 0110 does not
appear.
> I might be able to help if I were familiar with the
> terminology that you're using: what's your 
de[CapitalThorn]nition of a
> decomposition?
> [...]
> Brian
===
Subject: Re: Lagrange multipliers



 My son hit me with a calculus problem that I'm
having trouble
> getting started on. (its been 35 years since I had
my calculus and
> I don't even remember doing Lagrange multipliers but
I have an
idea
> of what they are doing).
 Anyway, the section they are in is Lagrange
multipliers and the
> problem is:
 Find the highest point on the curve of intersection
of the
surfaces
> sphere (x^2)+(y^2)+(z^2)=36 and
> plane 2x+y-z=2
 I'm having a hard time getting started. This 
doesn't
seem to [CapitalThorn]t
> the surface/constraint concept where you just take
the grad of the
> two surfaces (one with a constant multiplier to get
equal parallel
> vectors) and solve the simultaneous equations.
 Any hints? Do I need to pick a point outside the
sphere to work
> against or is this really a problem for Lagrange
multipliers? Did
> the book author throw in a ringer?

> tim
 Presuming the usual arrangement of xyz coordinates in
which
> increasing z is upward while changes in x and y are
horizontal, you
> want to mazimize z subject to restrictions
(x^2)+(y^2)+(z^2)-36 = 0
> and 2x+y-z-2=0
 Thus you have the Lagrange expression
> f(x,y,z,u,v) = z + u((x^2)+(y^2)+(z^2)-36) +
v(2x+y-z-2)
> with u and v as the multipliers.
 Ok, you are setting the equations as constraints
instead of
> surface/constraint. This would lead to the
> general form of : grad f(x,y,z) = (u)grad
g(x,y,z)+(v)grad h(x,y,z)
 This is where I get stuck. What do we use as function
f(x,y,z)? If
> this were a point and a plane, you could use the
equation of a sphere
> around the point with the plane as a constraint. In
this problem if
> you use the plane and the sphere as constraints, then
what do you
pick
> for the equation to maximize? A circle since that is
what a plane
> cutting a sphere should generate?
 tim
 Actually, you use grad(f((x,y,z,u,v)) = 0, where you
have a
> 5-dimensional function and its 5 dimensional gradient
> f(x,y,z,u,v) = z + u((x^2)+(y^2)+(z^2)-36) + v(2x+y-z-2)

> Letting f_w represent the partial with respect to w you
have
> f_x(x,y,z,u,v) = 2 u x + 2 v = 0
> f_y(x,y,z,u,v) = 2 u y + v = 0
> f_z(x,y,z,u,v) = 2 u z - v = 0
> f_u(x,y,z,u,v) = x^2 + y^2 + z^2 - 36 = 0
> f_w(x,y,z,u,v) = 2 x + y - z - 2 = 0
 I appreciate the help and I don't mean to quibble but
wouldn't
> f_z(x,y,z,u,v) = 1 + 2 u z - v = 0
> if f(x,y,z,u,v) = z + u z^2 - v z + ......... ?
> tim
> Right, My error. It often pays to proofread one more time!
Virgil,
equation in order to get the answer but it seems to have
worked.
I am impressed with the level of knowledge here. Keep up the
good work!
tim
===
Subject: Re: Lagrange multipliers





 My son hit me with a calculus problem that I'm
having trouble
> getting started on. (its been 35 years since I had
my calculus
and
> I don't even remember doing Lagrange multipliers
but I have an
idea
> of what they are doing).
 Anyway, the section they are in is Lagrange
multipliers and the
> problem is:
 Find the highest point on the curve of
intersection of the
surfaces
> sphere (x^2)+(y^2)+(z^2)=36 and
> plane 2x+y-z=2
 I'm having a hard time getting started. This
doesn't seem to [CapitalThorn]t
> the surface/constraint concept where you just take
the grad of
the
> two surfaces (one with a constant multiplier to
get equal
parallel
> vectors) and solve the simultaneous equations.
 Any hints? Do I need to pick a point outside the
sphere to work
> against or is this really a problem for Lagrange
multipliers?
Did
> the book author throw in a ringer?

> tim
 Presuming the usual arrangement of xyz coordinates
in which
> increasing z is upward while changes in x and y are
horizontal,
you
> want to mazimize z subject to restrictions
(x^2)+(y^2)+(z^2)-36 =
0
> and 2x+y-z-2=0
 Thus you have the Lagrange expression
> f(x,y,z,u,v) = z + u((x^2)+(y^2)+(z^2)-36) +
v(2x+y-z-2)
> with u and v as the multipliers.
 Ok, you are setting the equations as constraints
instead of
> surface/constraint. This would lead to the
> general form of : grad f(x,y,z) = (u)grad
g(x,y,z)+(v)grad
h(x,y,z)
 This is where I get stuck. What do we use as function
f(x,y,z)? If
> this were a point and a plane, you could use the
equation of a
sphere
> around the point with the plane as a constraint. In
this problem if
> you use the plane and the sphere as constraints, then
what do you
pick
> for the equation to maximize? A circle since that is
what a plane
> cutting a sphere should generate?
 tim
 Actually, you use grad(f((x,y,z,u,v)) = 0, where you
have a
> 5-dimensional function and its 5 dimensional gradient
> f(x,y,z,u,v) = z + u((x^2)+(y^2)+(z^2)-36) +
v(2x+y-z-2)

> Letting f_w represent the partial with respect to w
you have
> f_x(x,y,z,u,v) = 2 u x + 2 v = 0
> f_y(x,y,z,u,v) = 2 u y + v = 0
> f_z(x,y,z,u,v) = 2 u z - v = 0
> f_u(x,y,z,u,v) = x^2 + y^2 + z^2 - 36 = 0
> f_w(x,y,z,u,v) = 2 x + y - z - 2 = 0

> I appreciate the help and I don't mean to quibble but
wouldn't
> f_z(x,y,z,u,v) = 1 + 2 u z - v = 0
 if f(x,y,z,u,v) = z + u z^2 - v z + ......... ?
 tim
 Right, My error. It often pays to proofread one more time!
> Virgil,
> equation in order to get the answer but it seems to have
worked.
> I am impressed with the level of knowledge here. Keep up
the good work!
> tim
Glad to have been of use.
===
Subject: Re: Statistics
: I already tried to [CapitalThorn]nd the average and they are both
exactly the same
: (79.1). So I tried to see which had the most good numbers
but those
: seem to be Very close too...too close for me to tell which
is better.
: So I guess my question is: how on earth do I [CapitalThorn]gure this
out??
As I said in an earlier post, there is no clear answer unless
further assumptions are made. If you assume the expected
damage
is a linear function of the test score, both groups are
equally
good (if the goal is to minimize the expected damage). It the
expected damage is, for example, inversely proportional to the
test score, you would select the group with the lower average
reciprocal test score. There are many possibilities, and it is
not possible to proceed intelligently without a model relating
the score to the damage.
===
Subject: Periodic Functions - HELP
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9E69DV26291;
In my homework, there are several problems involving periodic
functions which I really don't understand, and would greatly
appreciate any help I could get! :-)
In observing a graph, (which I found the period to be .8, and
the
amplitude to be 1/2) I am then told to solve for f(1000) and
f(-1000).
Is there a simple equation to do this?
-Michelle
===
Subject: Re: Periodic Functions - HELP
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECcjq26804;
>In my homework, there are several problems involving periodic
>functions which I really don't understand, and would 
greatly
>appreciate any help I could get! :-)
>In observing a graph, (which I found the period to be .8,
and the
>amplitude to be 1/2) I am then told to solve for f(1000) and
f(-1000).
>Is there a simple equation to do this?
>-Michelle
Your function is periodic on 0.8 interval ,
Just consider 1 000 and -1 000 vs .8
integer multiple ? +/- 1250
Come on,Alain.
===
Subject: Re: Periodic Functions - HELP
> In my homework, there are several problems involving
periodic
> functions which I really don't understand, and would 
greatly
> appreciate any help I could get! :-)
> In observing a graph, (which I found the period to be .8,
and the
> amplitude to be 1/2) I am then told to solve for f(1000)
and f(-1000).
> Is there a simple equation to do this?
If the period is .8 then
f(1000) = f(1000 - 0.8) = f(1000 - 2 * 0.8) = f(1000 - 3 *
0.8)
...you get the idea.
hth
meeroh
--
If this message helped you, consider buying an item
from my wish list: <http://web.meeroh.org/wishlist>
===
Subject: Need help on this equation
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9E69DN26321;
(a b)^0.625
De = 1.3-------------
(a+b)^0.250
Let a be the length and b be the width of the dimension of an
retangular item.
===
Subject: Re: Need help on this equation
> (a b)^0.625
> De = 1.3-------------
> (a+b)^0.250
> Let a be the length and b be the width of the dimension of
an
> retangular item.
So what is your question?
===
Subject: Re: Need help on this equation
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECchw26690;
> (a b)^0.625
> De = 1.3-------------
> (a+b)^0.250
> Let a be the length and b be the width of the dimension of
an
> retangular item.
>So what is your question?
What is the solution to De?
===
Subject: Re: Need help on this equation
> (a b)^0.625
> De = 1.3-------------
> (a+b)^0.250
 Let a be the length and b be the width of the dimension
of an
> retangular item.
So what is your question?
> What is the solution to De?
Depends on the values of a and b as indicated in the equation.
===
Subject: Re: Need help on this equation
> (a b)^0.625
> De = 1.3-------------
> (a+b)^0.250
 Let a be the length and b be the width of the dimension
of an
> retangular item.
>What is the solution to De?
You've already written down the solution.
Perhaps if you copied the exact question from your textbook
...
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com
Fortunately, I live in the United States of America, where we
are
gradually coming to understand that nothing we do is ever our
fault, especially if it is really stupid. --Dave Barry
===
Subject: Factor Groups
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9E69DA26299;
So I am having some trouble with showing that there is some
homomorphism here that is one to one. . .
This is the problem: Let G be the group of nonzero real
numbers under
multiplication, and let N = {1,-1}. Prove that G/N is
isomorphic to
the positive reals under multiplication.
===
Subject: Re: Factor Groups
days. My association with the Department is that of an
alumnus.
>So I am having some trouble with showing that there is some
>homomorphism here that is one to one. . .
>This is the problem: Let G be the group of nonzero real
numbers under
>multiplication, and let N = {1,-1}. Prove that G/N is
isomorphic to
>the positive reals under multiplication.
Start with G, and map it to the positive reals; say, x|->|x|,
the
absolute value of x.
(i) Is this a group homomorphism?
(ii) What is the kernel?
(iii) What does the [CapitalThorn]rst isomorphism theorem say about
G/ker(f), when
f is a group homomorphism?
In general: when trying to prove that a group G modulo a
normal
subgroup K, G/K, is isomorphic to some other group H, usually
the
easiest way to do it is to de[CapitalThorn]ne a surjective homomorphism
from G to
H which has kernel equal to K, and invoke the isomorphism
theorem.
--
It's not denial. I'm just very selective 
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Factor Groups
> This is the problem: Let G be the group of nonzero real
numbers under
> multiplication, and let N = {1,-1}. Prove that G/N is
isomorphic to
> the positive reals under multiplication.
By the fundamental theorem of homomorphisms, as
P = { x in R | 0 < x }
is a multiplicative group and
f:G -> P, x -> |x|
is a surjective homomorphism with
ker f = { -1,1 }
where is there problem?
===
Subject: EXPECTATIONS OF A MATRIX
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECchO26717;
HI!
SORRY FOR MY IGNORANCE
BUT HOW CAN I COMPUTE THE EXPECTATION OF A MATRIX?
IN PARTICULAR:
IF:
MATRIX is a (m * n) matrix
E(...) is the expectation operator
in practical terms, how can I calculate E(MATRIX)?
===
Subject: Re: EXPECTATIONS OF A MATRIX
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EDW4W31781;
>HI!
>SORRY FOR MY IGNORANCE
>BUT HOW CAN I COMPUTE THE EXPECTATION OF A MATRIX?
>IN PARTICULAR:
>IF:
>MATRIX is a (m * n) matrix
> E(...) is the expectation operator
>in practical terms, how can I calculate E(MATRIX)?
You simply [CapitalThorn]nd the expectation of each element. Let X =
[x_ij] be
the random matrix in question. Then its expected value M =
E(X) is
just the matrix of expected values of the elements of X:
M = [ E(x_ij) ]
Of course, you need to know the probability density function
(pdf),
p_ij, of each element.
The only problem with the answer above is that M may not be
the same
kind of matrix as X. For example, suppose X is a member of
SO(n), the
special orthogonal matrices ( (X)(X^t) = (X^t)(X) = I and
det(X) =
+1). M is, in general, not a member of SO(n). To make the
example
more speci[CapitalThorn]c, suppose X is a member of SO(2), and that it is
uniformly distributed on SO(2). Then X can be represented by
X =
[ cos(theta) -sin(theta) ]
[ sin(theta) cos(theta) ]
where theta is uniformly distributed on the interval [0,
2pi]. It
should be easy to see that M = E(X) = 0^(2 x 2) (the 2 x 2
matrix of
zeros), which is not a member of SO(n).
If you want to restrict M to be the same type of matrix as X,
then you
need to introduce your own method of taking expectations. To
my
knowledge (which is not necessarily that broad) there is no
standard
method for de[CapitalThorn]ning such an expectation.
-MO
===
Subject: Re: EXPECTATIONS OF A MATRIX
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EFdtl11282;
>HI!
>SORRY FOR MY IGNORANCE
>BUT HOW CAN I COMPUTE THE EXPECTATION OF A MATRIX?
>IN PARTICULAR:
>IF:
>MATRIX is a (m * n) matrix
> E(...) is the expectation operator
>in practical terms, how can I calculate E(MATRIX)?
>You simply [CapitalThorn]nd the expectation of each element. Let X =
[x_ij] be
>the random matrix in question. Then its expected value M =
E(X) is
>just the matrix of expected values of the elements of X:
>M = [ E(x_ij) ]
>Of course, you need to know the probability density function
(pdf),
>p_ij, of each element.
>The only problem with the answer above is that M may not be
the same
>kind of matrix as X. For example, suppose X is a member of
SO(n),
the
>special orthogonal matrices ( (X)(X^t) = (X^t)(X) = I and
det(X) =
>+1). M is, in general, not a member of SO(n). To make the
example
>more speci[CapitalThorn]c, suppose X is a member of SO(2), and that it is
>uniformly distributed on SO(2). Then X can be represented by
>X =
>[ cos(theta) -sin(theta) ]
>[ sin(theta) cos(theta) ]
>where theta is uniformly distributed on the interval [0,
2pi]. It
>should be easy to see that M = E(X) = 0^(2 x 2) (the 2 x 2
matrix of
>zeros), which is not a member of SO(n).
>If you want to restrict M to be the same type of matrix as
X, then
you
>need to introduce your own method of taking expectations. To
my
>knowledge (which is not necessarily that broad) there is no
standard
>method for de[CapitalThorn]ning such an expectation.
>-MO
in the basic linear regression case (homoscedastic errors)
how can I calculate the Expectation of the Matrix (n x n) of
the
residual terms, named variance-covariance matrix of the error
terms?
I could only assume that it is:
var(res) x I
where: I is the identity matrix
and var(res) is a scalar : the variance of the error term
===
Subject: Re: Another problem on Stokes's Theorem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECcij26769;
Anyone with any suggestons about how to solve this problem?
Michelle
===
Subject: Governing Equation and Boundary Conditon for Polar
orthotropic
circular plate.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECckL26863;
Can you provide me the governing equation and boundary
condition of
polar orthotropic cicular plates..Pls. send me the
formulation.
===
Subject: Governing Equation and Boundary Conditon for Polar
orthotropic
circular plate.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECckG26851;
Can you provide me the governing equation and boundary
condition of
polar orthotropic cicular plates..Pls. send me the
formulation.
===
Subject: Phrase Structure
Can someone please solve the following
Construct phrase-structure grammars to generate the following
set
{0^n1^2n | n >= 0}
===
Subject: Re: Phrase Structure
alt.math.undergrad:
> Can someone please solve the following
> Construct phrase-structure grammars to generate the
following set
> {0^n1^2n | n >= 0}
(I assume that this should be {0^n 1^(2n) : n >= 0}.) This
is really extremely straightforward; you should be able to
do it yourself. Here are a few hints, though. You can do
it with a single non-terminal symbol; the main production
should generate one new 0 and two new 1's. Have you seen a
grammar that generates {0^n 1^n : n >= 0}? If so, just
modify it slightly.
Brian
===
Subject: Truth Tables Help
I have a four part question in a tutorial, parts 1 & 2 I
don't have any
problem with. Parts 3 & 4 I have no idea what they mean. Can
someone
explain
to me what parts 3 & 4 mean and how can I answer the
questions.
Alternatively if you can point me to a website that might be
of some help.
The question:
Draw the truth tables for each of the following expressions:
1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
2. (A + B' + C' + D)(A' + B + C 
+ D')
3. Sm(2.5.6.9)
4. ?M(0,7,10,11)
Appreciate any help.
TIA
===
Subject: Truth Tables Help
My earlier post should read:
I have a four part question in a tutorial, parts 1 & 2 I
don't have any
problem with. Parts 3 & 4 I have no idea what they mean. Can
someone
explain
to me what parts 3 & 4 mean and how can I answer the
questions.
Alternatively if you can point me to a website that might be
of some help.
The question:
Draw the truth tables for each of the following expressions:
1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
2. (A + B' + C' + D)(A' + B + C 
+ D')
3. Sigma m(2.5.6.9)
4. Pi M(0,7,10,11)
3 & 4 should read:
_
/_ m (2.5.6.9)
__
I I M (0,7,10,11)
Appreciate any help.
TIA
===
Subject: Re: Truth Tables Help
> My earlier post should read:
> I have a four part question in a tutorial, parts 1 & 2 I
don't have any
> problem with. Parts 3 & 4 I have no idea what they mean.
Can someone
explain
> to me what parts 3 & 4 mean and how can I answer the
questions.
> Alternatively if you can point me to a website that might
be of some
help.
> The question:
> Draw the truth tables for each of the following expressions:
> 1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
> 2. (A + B' + C' + D)(A' + B + 
C + D')
> 3. Sigma m(2.5.6.9)
> 4. Pi M(0,7,10,11)
> 3 & 4 should read:
> /_ m (2.5.6.9)
> __
> I I M (0,7,10,11)
What do your texts or lecture notes say these notations mean?
Guess:
Sigma indicates + over some set, and Pi indicates . over some
set--but
what sets?
===
Subject: numeric series question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EJJV501477;
We know that:
inf
---
n+1 1 1 1 1
(-1) ----- = 1 - --- + --- - --- + ... = eta(1) = ln2
/ n 2 3 4
---
n=1
from the expansion of ln(1+x);
we also know that:
inf
---
n+1 1 1 1 1
(-1) ----- = 1 - --- + --- - --- + ... = pi/4
/ 2n-1 3 5 7
---
n=1
from the expansion of atan(x);
but what about:
inf
---
n+1 1 1 1 1
(-1) ----- = 1 - --- + --- - --- + ... = ???
/ 3n-2 4 7 10
---
n=1
more generally,
inf
---
n+1 1
(-1) -------- = ??? [k integer >=1] (A)
/ kn-(k-1)
---
n=1
It's interesting to note that the family of alternating
series (A)
above,
is connected with
inf
---
1 1
-------------- = --- [for every integer k>=1] (B)
/ (kn-k+1)(kn+1) k
---
n=1
In order to see how, take k=1: the series (B) is: inf
---
1 1 1 1 1 1 1
----- + ----- + ----- + ----- + ----- + ----- + ... = -------
= 1
1*2 2*3 3*4 4*5 5*6 6*7 / n(n+1)
---
n=1
Now the odd terms make up the series (A) with k=1:
inf
---
1 1 1 1 n+1 1
----- + ----- + ----- + ----- + ... = (-1) --- = ln2
1*2 3*4 5*6 6*7 / n
---
n=1
because (1 - 1/2) = 1/2; (1/3 - 1/4) = 1/12 etc.
(1/n - 1/(n+1)) = 1/n*(n+1)
The even terms make up another subseries, with sum = 1-ln2
For k>1 there is a coef[CapitalThorn]cient (which is exactly 1/k) to take
out...
eg. k=2 we want to know the Ôodd' subseries of 
(B),
1/(1*3) + 1/(5*7) + 1/(7*9) + ...
well,
1 1 1
1 - --- --- - ---
1 3 1 5 7
----- = --------- , ----- = ----------- , etc.
1*3 2 5*7 2
because we know that
1-(1/3)+(1/5)-(1/7)+ etc. = pi/4,
our subseries is pi/8 !
In the end, for k=1, series (B) is split in two series, one
with sum =
ln2,
another with sum = 1-ln2.
For k=2 the similar split gives a series with sum = pi/8 and
another
with sum
(1/2 - pi/8).
What is the Ôsplit' for k>=3, k integer?
That is, which is the sum of the Ôodd' subseries 
of (B), for k
integer, k>=3?
In order to calculate this you need to know the sum of (A)
for k>=3,
but,
how much is it?
Is it known or is it still unknown in its concise form, like
zeta(2k+1)?
===
Subject: Re: numeric series question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ELLBc12432;
Here's a hint.
Use the expression for ln(1+x) where x=i, and where x=-i.
See if you can combine the series for ln(1+i) and for ln(1-i)
to get the alternating sum
1/1 - 1/3 + 1/5 - 1/7 + 1/9 -+ ... .
(Subtract the two series to get some unwanted terms to
cancel.)
Notice this is the same route that we use to get
1/1 - 1/2 + 1/3 - 1/4 + 1/5 -+ ... .
For the series you asked about,
1/1 - 1/4 + 1/7 - 1/10 + 1/13 - 1/16 +- ...
you want the coef[CapitalThorn]cients of 1/n to be periodic with period 6:
1,0,0,-1,0,0,1,0,0,-1,0,0,1,...
This will happen if you look at ln(1+x)
where x is a sixth root of unity.
You have six choices here: if w=exp(2 pi i / 6),
we can let x=w, w^2, w^3=-1, w^4, w^5, or w^6=1.
(The last one doesn't help.)
Write out those six series, and see whether you can [CapitalThorn]nd some
linear combination of them that gives the series you want.
Then generalize to the other values of k.
===
Subject: Descrete Math help with question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ELLBD12436;
I'v been trying to [CapitalThorn]gure out this question and 
i don't even
know
where to begin.
De[CapitalThorn]ne two Sets A and B as follows: A = {(2n+1)^3|n E Z} and
B = {(2n+1)|n E Z}
a) prove that A is a proper subset of B.
b) suppose we redi[CapitalThorn]ne A and B, replacing Z by R . What is the
relation between these two sets? State and prove your answer.
===
Subject: Re: Descrete Math help with question
days. My association with the Department is that of an
alumnus.
>I'v been trying to [CapitalThorn]gure out this question and 
i don't even
know
>where to begin.
>De[CapitalThorn]ne two Sets A and B as follows: A = {(2n+1)^3|n E Z} and
>B = {(2n+1)|n E Z}
>a) prove that A is a proper subset of B.
The [CapitalThorn]rst set, A, contains the cubes of the odd integers. So 
it
contains 1 (=1^3), -1 (=-1^3), 27 (=3^3), -27 (=(-3)^3), 125,
-125,
etc.
Set B contains the odd integers.
So: can you show that
(a) every element of A is an element of B (i.e., A is
contained in
B)?
For example, take an arbitrary element of A, (2k+1)^3 = 8k^3 +
12k^2 + 6k + 1. Can you should that it can be written as 2n+1
for
some integer n? If so, then it would necessarily be in B.
(b) That there is at least one element of B which is not in A
(so A
is not equal to B)? For example: when n=1, you get the element
(2+1)=3 in B. Can you show that there is NO integer k such
that
(2k+1)^3 = 3?
>b) suppose we redi[CapitalThorn]ne A and B, replacing Z by R . What is 
the
>relation between these two sets? State and prove your answer.
So now A contains all real numbers which can be expressed as
(2y+1)^3,
and B contains all real numbers which can be expressed as
(2x+1), with
x and y any real numbers.
IS there any relation between A and B now? Is A contained in
B? Is A
properly contained in B? Is B contained in A? Is B properly
contained
in B?
--
It's not denial. I'm just very selective 
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu

===
Subject: Regular Sets
I am totally lost on this one...can someone please help...
Determine whether 1011 belongs to each of these regular sets:
10*1*
0*(10 U 11)*
1(01)*1*
1*01(0 U 1)
(10)*(11)*
1(00)*(11)*
(10)*1011
(1 U 00)(01 U 0)1*
===
Subject: Re: Regular Sets
alt.math.undergrad:
> I am totally lost on this one...can someone please help...
> Determine whether 1011 belongs to each of these regular
sets:
In each case you have to decide whether the string 1011 can
be made to match the indicated pattern. To do this you have
to be able to interpret the notation for regular sets. I'll
do the [CapitalThorn]rst three in some detail and let you try your hand
again on the rest.
> 10*1*
What has to be true of a string in order for it to match
this pattern? It must:
(a) begin with a 1;
(b) then have zero or more 0's; and
(c) [CapitalThorn]nish with zero or more 1's.
Since you can break up 1011 as
1 0 11
(a) (b) (c),
it matches this pattern and does belong to the set.
> 0*(10 U 11)*
This one's a little more complicated. It describes strings
that can be broken into two parts, say uv, where u is in 0*
and v is in (10 U 11)*. Can you split 1011 in such a way?
0* includes only the empty string, 0, 00, 000, 0000, and so
on; since 1011 begins with a 1, the only possible u is the
empty string, so we'll try it. That means that the v will
have to be 1011 itself; is that in (10 U 11)*? To answer
that, you have to understand what's in (10 U 11)*. The
empty string is, but what else there? 10 and 11 are. So
are 1010, 1011, 1110, and 1111. And in general so is any
string that you can build by repeatedly choosing one of 10
and 11 and appending it to one that you already have. The
ones of length 6, for instance, are 101010, 101011, 101110,
101111, 111010, 111011, 111110, and 111111; each of these is
one of the strings of length 4 extended by either 10 or 11
on the right.
In particular, 1011 is in (10 U 11)*, so it's also in
0*(10 U 11)*, with the invisible empty string at its front
matching 0*:
0* (10 U 11)*
- 10 11
> 1(01)*1*
The only way to match a pattern 1 is with the string 1, so
if 1011 is going to match this pattern, it has to be because
the [CapitalThorn]rst 1 matches 1 and the remaining 011 matches (01)*1*.
This means that 011 must decompose into two substrings, say
011 = uv, such that u matches (01)* and v matches 1*. This
is easy: take u = 01 and v = 1. (Remember, (01)* consists
of the empty string, 01, 0101, 010101, and so on: strings of
zero or more copies of 01. Similarly, 1* consists of
strings of zero or more 1's.) Thus, 1011 matches 1(01)*1*
as set out below:
1 (10)* 1*
1 10 11
One hint for the remaining [CapitalThorn]ve: 1011 is in all but one of
the sets.
> 1*01(0 U 1)
> (10)*(11)*
> 1(00)*(11)*
> (10)*1011
> (1 U 00)(01 U 0)1*
Brian
===
Subject: Re: Uniform Convergence
alt.math.undergrad:
[...]
> I am trying to develop logarithms in a different way
> from the classical way of considering the integral of
> 1/x. So I differentiate c^x with respect to x, and get
> the limit as h tends to 0 of f(c,h) multiplied by c^x.
> Assuming you seek rigor here, it seems to me you have
> some problems. What is the de[CapitalThorn]nition of c^x? Why is it
> differentiable?
[...]
> De[CapitalThorn]ne c^x in the standard way as follows:
> Suppose x is a natural number, then c^x is de[CapitalThorn]ned
> inductively as c^1 = c and c^(x+1)=c*c^x (the proof that
> this is well de[CapitalThorn]ned follows from the principle of
> induction trivially).
> Suppose x is an integer. If x is 0, c^x is 1. If x is
> negative, c^x is 1/c^(-x).
> Suppose x is rational. Then x = p/q, p and q integers and
> q non-zero. Then c^x is the solution y of the equation
> y^q = c^p.
To make this rigorous you have to show that if p/q = r/s,
then y^q = c^p and y^s = c^r have the same solutions. You
also have to deal with the fact that y^2 = c^1 has two
solutions (for c > 0, which is presumably the only case that
you're considering). You also have to check that 
it's
consistent with the earlier parts of the de[CapitalThorn]nition, though
that's pretty trivial.
> Suppose x is real. Then there is a sequence of rationals
> x_n tending to x. So c^x is the limit as n tends to
> in[CapitalThorn]nite of c^(x_n).
To make this rigorous you have to prove that the limit
exists and is independent of the particular sequence <x_n>
chosen. You also have to show that this is consistent with
the earlier parts of the de[CapitalThorn]nition.
> Suppose x is complex. Then c^x is exp(x*ln(c)).
How are you de[CapitalThorn]ning the logarithm on C? I thought that
this was in aid of developing the logarithm function. Or
are you just saying that once you have real logs, you'll
develop complex logs in the usual way?
[...]
Brian
===
Subject: Re: Uniform Convergence
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EG4AI14235;
>alt.math.undergrad:
>[...]
> I am trying to develop logarithms in a different way
> from the classical way of considering the integral of
> 1/x. So I differentiate c^x with respect to x, and get
> the limit as h tends to 0 of f(c,h) multiplied by c^x.
> Assuming you seek rigor here, it seems to me you have
> some problems. What is the de[CapitalThorn]nition of c^x? Why is it
> differentiable?
>[...]
> De[CapitalThorn]ne c^x in the standard way as follows:
> Suppose x is a natural number, then c^x is de[CapitalThorn]ned
> inductively as c^1 = c and c^(x+1)=c*c^x (the proof that
> this is well de[CapitalThorn]ned follows from the principle of
> induction trivially).
> Suppose x is an integer. If x is 0, c^x is 1. If x is
> negative, c^x is 1/c^(-x).
> Suppose x is rational. Then x = p/q, p and q integers and
> q non-zero. Then c^x is the solution y of the equation
> y^q = c^p.
>To make this rigorous you have to show that if p/q = r/s,
>then y^q = c^p and y^s = c^r have the same solutions. You
>also have to deal with the fact that y^2 = c^1 has two
>solutions (for c > 0, which is presumably the only case that
>you're considering). You also have to check that 
it's
>consistent with the earlier parts of the de[CapitalThorn]nition, though
>that's pretty trivial.
> Suppose x is real. Then there is a sequence of rationals
> x_n tending to x. So c^x is the limit as n tends to
> in[CapitalThorn]nite of c^(x_n).
>To make this rigorous you have to prove that the limit
>exists and is independent of the particular sequence <x_nchosen. You also have to show that this is consistent with
>the earlier parts of the de[CapitalThorn]nition.
> Suppose x is complex. Then c^x is exp(x*ln(c)).
>How are you de[CapitalThorn]ning the logarithm on C? I thought that
>this was in aid of developing the logarithm function. Or
>are you just saying that once you have real logs, you'll
>develop complex logs in the usual way?
>[...]
>Brian
Oh dear, I hadn't really thought of that last point. But
yeah, I'll do
that - I'll develop ln for real argument and then 
de[CapitalThorn]ne c^x
where c
is real and x complex.
In terms of your earlier points - I had thought of most of
these, but
what do you mean for real numbers where you say you need to
prove that
the limit exists? I am de[CapitalThorn]ning x_n as a rational sequence
converging
to x, so obviously the limit exists, otherwise the sequence
does not
satisfy the de[CapitalThorn]nition!
===
Subject: Re: Uniform Convergence
alt.math.undergrad:
>alt.math.undergrad:
>[...]
> I am trying to develop logarithms in a different way
> from the classical way of considering the integral of
> 1/x. So I differentiate c^x with respect to x, and get
> the limit as h tends to 0 of f(c,h) multiplied by c^x.
> Assuming you seek rigor here, it seems to me you have
> some problems. What is the de[CapitalThorn]nition of c^x? Why is it
> differentiable?
>[...]
> De[CapitalThorn]ne c^x in the standard way as follows:
> Suppose x is a natural number, then c^x is de[CapitalThorn]ned
> inductively as c^1 = c and c^(x+1)=c*c^x (the proof that
> this is well de[CapitalThorn]ned follows from the principle of
> induction trivially).
> Suppose x is an integer. If x is 0, c^x is 1. If x is
> negative, c^x is 1/c^(-x).
> Suppose x is rational. Then x = p/q, p and q integers and
> q non-zero. Then c^x is the solution y of the equation
> y^q = c^p.
>To make this rigorous you have to show that if p/q = r/s,
>then y^q = c^p and y^s = c^r have the same solutions. You
>also have to deal with the fact that y^2 = c^1 has two
>solutions (for c > 0, which is presumably the only case that
>you're considering). You also have to check that 
it's
>consistent with the earlier parts of the de[CapitalThorn]nition, though
>that's pretty trivial.
> Suppose x is real. Then there is a sequence of rationals
> x_n tending to x. So c^x is the limit as n tends to
> in[CapitalThorn]nite of c^(x_n).
>To make this rigorous you have to prove that the limit
>exists and is independent of the particular sequence <x_nchosen. You also have to show that this is consistent with
>the earlier parts of the de[CapitalThorn]nition.
[...]
> In terms of your earlier points - I had thought of most of
these, but
> what do you mean for real numbers where you say you need to
prove that
> the limit exists?
You have to prove that if <x_n : n in N> converges to x,
then <c^{x_n} : n in N> actually converges. This is not
trivial.
[...]
Brian
===
Subject: Decomposition of {0,1} strings problem.
I can't [CapitalThorn]gure this one out.
I need to write a decomposition for all {0,1} strings that
uniquely
creates all strings such that the substring 0110 does not
appear.
Now, I was able to get 011, but I'm unable to 
[CapitalThorn]gure out a
decomposition
for 0110. I came up with a bunch, but none of they are
uniquely created.
Can anybody offer any hints as to how I could do this?
===
Subject: Re: Decomposition of {0,1} strings problem.
alt.math.undergrad:
> I can't [CapitalThorn]gure this one out.
> I need to write a decomposition for all {0,1} strings that
uniquely
> creates all strings such that the substring 0110 does not
appear.
I might be able to help if I were familiar with the
terminology that you're using: what's your 
de[CapitalThorn]nition of a
decomposition?
[...]
Brian
===
Subject: Re: Decomposition of {0,1} strings problem.
For example, so far I have the following:
{1}*({0}({E,1}U{111}{1}*))* and that seems very close, but it
doesn't
account for 011. I can't just add {1}* at the end since it
would no
longer be uniquely created.
> alt.math.undergrad:
>I can't [CapitalThorn]gure this one out.
>I need to write a decomposition for all {0,1} strings that
uniquely
>creates all strings such that the substring 0110 does not
appear.
> I might be able to help if I were familiar with the
> terminology that you're using: what's your 
de[CapitalThorn]nition of a
> decomposition?
> [...]
> Brian
===
Subject: Re: Decomposition of {0,1} strings problem.
Nevermind, I [CapitalThorn]gured it out
You need to add {E,011} to the end
> For example, so far I have the following:
> {1}*({0}({E,1}U{111}{1}*))* and that seems very close, but
it doesn't
> account for 011. I can't just add {1}* at the end since it
would no
> longer be uniquely created.
> alt.math.undergrad:
> I can't [CapitalThorn]gure this one out.
> I need to write a decomposition for all {0,1} strings
that uniquely
> creates all strings such that the substring 0110 does not
appear.
> I might be able to help if I were familiar with the
> terminology that you're using: what's your 
de[CapitalThorn]nition of a
> decomposition?
> [...]
> Brian
===
Subject: Re: Lagrange multipliers



 My son hit me with a calculus problem that I'm
having trouble
> getting started on. (its been 35 years since I had
my calculus and
> I don't even remember doing Lagrange multipliers but
I have an
idea
> of what they are doing).
 Anyway, the section they are in is Lagrange
multipliers and the
> problem is:
 Find the highest point on the curve of intersection
of the
surfaces
> sphere (x^2)+(y^2)+(z^2)=36 and
> plane 2x+y-z=2
 I'm having a hard time getting started. This 
doesn't
seem to [CapitalThorn]t
> the surface/constraint concept where you just take
the grad of the
> two surfaces (one with a constant multiplier to get
equal parallel
> vectors) and solve the simultaneous equations.
 Any hints? Do I need to pick a point outside the
sphere to work
> against or is this really a problem for Lagrange
multipliers? Did
> the book author throw in a ringer?

> tim
 Presuming the usual arrangement of xyz coordinates in
which
> increasing z is upward while changes in x and y are
horizontal, you
> want to mazimize z subject to restrictions
(x^2)+(y^2)+(z^2)-36 = 0
> and 2x+y-z-2=0
 Thus you have the Lagrange expression
> f(x,y,z,u,v) = z + u((x^2)+(y^2)+(z^2)-36) +
v(2x+y-z-2)
> with u and v as the multipliers.
 Ok, you are setting the equations as constraints
instead of
> surface/constraint. This would lead to the
> general form of : grad f(x,y,z) = (u)grad
g(x,y,z)+(v)grad h(x,y,z)
 This is where I get stuck. What do we use as function
f(x,y,z)? If
> this were a point and a plane, you could use the
equation of a sphere
> around the point with the plane as a constraint. In
this problem if
> you use the plane and the sphere as constraints, then
what do you
pick
> for the equation to maximize? A circle since that is
what a plane
> cutting a sphere should generate?
 tim
 Actually, you use grad(f((x,y,z,u,v)) = 0, where you
have a
> 5-dimensional function and its 5 dimensional gradient
> f(x,y,z,u,v) = z + u((x^2)+(y^2)+(z^2)-36) + v(2x+y-z-2)

> Letting f_w represent the partial with respect to w you
have
> f_x(x,y,z,u,v) = 2 u x + 2 v = 0
> f_y(x,y,z,u,v) = 2 u y + v = 0
> f_z(x,y,z,u,v) = 2 u z - v = 0
> f_u(x,y,z,u,v) = x^2 + y^2 + z^2 - 36 = 0
> f_w(x,y,z,u,v) = 2 x + y - z - 2 = 0
 I appreciate the help and I don't mean to quibble but
wouldn't
> f_z(x,y,z,u,v) = 1 + 2 u z - v = 0
> if f(x,y,z,u,v) = z + u z^2 - v z + ......... ?
> tim
> Right, My error. It often pays to proofread one more time!
Virgil,
equation in order to get the answer but it seems to have
worked.
I am impressed with the level of knowledge here. Keep up the
good work!
tim
===
Subject: Re: Lagrange multipliers





 My son hit me with a calculus problem that I'm
having trouble
> getting started on. (its been 35 years since I had
my calculus
and
> I don't even remember doing Lagrange multipliers
but I have an
idea
> of what they are doing).
 Anyway, the section they are in is Lagrange
multipliers and the
> problem is:
 Find the highest point on the curve of
intersection of the
surfaces
> sphere (x^2)+(y^2)+(z^2)=36 and
> plane 2x+y-z=2
 I'm having a hard time getting started. This
doesn't seem to [CapitalThorn]t
> the surface/constraint concept where you just take
the grad of
the
> two surfaces (one with a constant multiplier to
get equal
parallel
> vectors) and solve the simultaneous equations.
 Any hints? Do I need to pick a point outside the
sphere to work
> against or is this really a problem for Lagrange
multipliers?
Did
> the book author throw in a ringer?

> tim
 Presuming the usual arrangement of xyz coordinates
in which
> increasing z is upward while changes in x and y are
horizontal,
you
> want to mazimize z subject to restrictions
(x^2)+(y^2)+(z^2)-36 =
0
> and 2x+y-z-2=0
 Thus you have the Lagrange expression
> f(x,y,z,u,v) = z + u((x^2)+(y^2)+(z^2)-36) +
v(2x+y-z-2)
> with u and v as the multipliers.
 Ok, you are setting the equations as constraints
instead of
> surface/constraint. This would lead to the
> general form of : grad f(x,y,z) = (u)grad
g(x,y,z)+(v)grad
h(x,y,z)
 This is where I get stuck. What do we use as function
f(x,y,z)? If
> this were a point and a plane, you could use the
equation of a
sphere
> around the point with the plane as a constraint. In
this problem if
> you use the plane and the sphere as constraints, then
what do you
pick
> for the equation to maximize? A circle since that is
what a plane
> cutting a sphere should generate?
 tim
 Actually, you use grad(f((x,y,z,u,v)) = 0, where you
have a
> 5-dimensional function and its 5 dimensional gradient
> f(x,y,z,u,v) = z + u((x^2)+(y^2)+(z^2)-36) +
v(2x+y-z-2)

> Letting f_w represent the partial with respect to w
you have
> f_x(x,y,z,u,v) = 2 u x + 2 v = 0
> f_y(x,y,z,u,v) = 2 u y + v = 0
> f_z(x,y,z,u,v) = 2 u z - v = 0
> f_u(x,y,z,u,v) = x^2 + y^2 + z^2 - 36 = 0
> f_w(x,y,z,u,v) = 2 x + y - z - 2 = 0

> I appreciate the help and I don't mean to quibble but
wouldn't
> f_z(x,y,z,u,v) = 1 + 2 u z - v = 0
 if f(x,y,z,u,v) = z + u z^2 - v z + ......... ?
 tim
 Right, My error. It often pays to proofread one more time!
> Virgil,
> equation in order to get the answer but it seems to have
worked.
> I am impressed with the level of knowledge here. Keep up
the good work!
> tim
Glad to have been of use.
===
Subject: Re: Statistics
: I already tried to [CapitalThorn]nd the average and they are both
exactly the same
: (79.1). So I tried to see which had the most good numbers
but those
: seem to be Very close too...too close for me to tell which
is better.
: So I guess my question is: how on earth do I [CapitalThorn]gure this
out??
As I said in an earlier post, there is no clear answer unless
further assumptions are made. If you assume the expected
damage
is a linear function of the test score, both groups are
equally
good (if the goal is to minimize the expected damage). It the
expected damage is, for example, inversely proportional to the
test score, you would select the group with the lower average
reciprocal test score. There are many possibilities, and it is
not possible to proceed intelligently without a model relating
the score to the damage.
===
Subject: Periodic Functions - HELP
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9E69DV26291;
In my homework, there are several problems involving periodic
functions which I really don't understand, and would greatly
appreciate any help I could get! :-)
In observing a graph, (which I found the period to be .8, and
the
amplitude to be 1/2) I am then told to solve for f(1000) and
f(-1000).
Is there a simple equation to do this?
-Michelle
===
Subject: Re: Periodic Functions - HELP
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECcjq26804;
>In my homework, there are several problems involving periodic
>functions which I really don't understand, and would 
greatly
>appreciate any help I could get! :-)
>In observing a graph, (which I found the period to be .8,
and the
>amplitude to be 1/2) I am then told to solve for f(1000) and
f(-1000).
>Is there a simple equation to do this?
>-Michelle
Your function is periodic on 0.8 interval ,
Just consider 1 000 and -1 000 vs .8
integer multiple ? +/- 1250
Come on,Alain.
===
Subject: Re: Periodic Functions - HELP
> In my homework, there are several problems involving
periodic
> functions which I really don't understand, and would 
greatly
> appreciate any help I could get! :-)
> In observing a graph, (which I found the period to be .8,
and the
> amplitude to be 1/2) I am then told to solve for f(1000)
and f(-1000).
> Is there a simple equation to do this?
If the period is .8 then
f(1000) = f(1000 - 0.8) = f(1000 - 2 * 0.8) = f(1000 - 3 *
0.8)
...you get the idea.
hth
meeroh
--
If this message helped you, consider buying an item
from my wish list: <http://web.meeroh.org/wishlist>
===
Subject: Need help on this equation
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9E69DN26321;
(a b)^0.625
De = 1.3-------------
(a+b)^0.250
Let a be the length and b be the width of the dimension of an
retangular item.
===
Subject: Re: Need help on this equation
> (a b)^0.625
> De = 1.3-------------
> (a+b)^0.250
> Let a be the length and b be the width of the dimension of
an
> retangular item.
So what is your question?
===
Subject: Re: Need help on this equation
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECchw26690;
> (a b)^0.625
> De = 1.3-------------
> (a+b)^0.250
> Let a be the length and b be the width of the dimension of
an
> retangular item.
>So what is your question?
What is the solution to De?
===
Subject: Re: Need help on this equation
> (a b)^0.625
> De = 1.3-------------
> (a+b)^0.250
 Let a be the length and b be the width of the dimension
of an
> retangular item.
So what is your question?
> What is the solution to De?
Depends on the values of a and b as indicated in the equation.
===
Subject: Re: Need help on this equation
> (a b)^0.625
> De = 1.3-------------
> (a+b)^0.250
 Let a be the length and b be the width of the dimension
of an
> retangular item.
>What is the solution to De?
You've already written down the solution.
Perhaps if you copied the exact question from your textbook
...
--
Stan Brown, Oak Road Systems, Tompkins County, New York, USA
http://OakRoadSystems.com
Fortunately, I live in the United States of America, where we
are
gradually coming to understand that nothing we do is ever our
fault, especially if it is really stupid. --Dave Barry
===
Subject: Factor Groups
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9E69DA26299;
So I am having some trouble with showing that there is some
homomorphism here that is one to one. . .
This is the problem: Let G be the group of nonzero real
numbers under
multiplication, and let N = {1,-1}. Prove that G/N is
isomorphic to
the positive reals under multiplication.
===
Subject: Re: Factor Groups
days. My association with the Department is that of an
alumnus.
>So I am having some trouble with showing that there is some
>homomorphism here that is one to one. . .
>This is the problem: Let G be the group of nonzero real
numbers under
>multiplication, and let N = {1,-1}. Prove that G/N is
isomorphic to
>the positive reals under multiplication.
Start with G, and map it to the positive reals; say, x|->|x|,
the
absolute value of x.
(i) Is this a group homomorphism?
(ii) What is the kernel?
(iii) What does the [CapitalThorn]rst isomorphism theorem say about
G/ker(f), when
f is a group homomorphism?
In general: when trying to prove that a group G modulo a
normal
subgroup K, G/K, is isomorphic to some other group H, usually
the
easiest way to do it is to de[CapitalThorn]ne a surjective homomorphism
from G to
H which has kernel equal to K, and invoke the isomorphism
theorem.
--
It's not denial. I'm just very selective 
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu
===
Subject: Re: Factor Groups
> This is the problem: Let G be the group of nonzero real
numbers under
> multiplication, and let N = {1,-1}. Prove that G/N is
isomorphic to
> the positive reals under multiplication.
By the fundamental theorem of homomorphisms, as
P = { x in R | 0 < x }
is a multiplicative group and
f:G -> P, x -> |x|
is a surjective homomorphism with
ker f = { -1,1 }
where is there problem?
===
Subject: EXPECTATIONS OF A MATRIX
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECchO26717;
HI!
SORRY FOR MY IGNORANCE
BUT HOW CAN I COMPUTE THE EXPECTATION OF A MATRIX?
IN PARTICULAR:
IF:
MATRIX is a (m * n) matrix
E(...) is the expectation operator
in practical terms, how can I calculate E(MATRIX)?
===
Subject: Re: EXPECTATIONS OF A MATRIX
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EDW4W31781;
>HI!
>SORRY FOR MY IGNORANCE
>BUT HOW CAN I COMPUTE THE EXPECTATION OF A MATRIX?
>IN PARTICULAR:
>IF:
>MATRIX is a (m * n) matrix
> E(...) is the expectation operator
>in practical terms, how can I calculate E(MATRIX)?
You simply [CapitalThorn]nd the expectation of each element. Let X =
[x_ij] be
the random matrix in question. Then its expected value M =
E(X) is
just the matrix of expected values of the elements of X:
M = [ E(x_ij) ]
Of course, you need to know the probability density function
(pdf),
p_ij, of each element.
The only problem with the answer above is that M may not be
the same
kind of matrix as X. For example, suppose X is a member of
SO(n), the
special orthogonal matrices ( (X)(X^t) = (X^t)(X) = I and
det(X) =
+1). M is, in general, not a member of SO(n). To make the
example
more speci[CapitalThorn]c, suppose X is a member of SO(2), and that it is
uniformly distributed on SO(2). Then X can be represented by
X =
[ cos(theta) -sin(theta) ]
[ sin(theta) cos(theta) ]
where theta is uniformly distributed on the interval [0,
2pi]. It
should be easy to see that M = E(X) = 0^(2 x 2) (the 2 x 2
matrix of
zeros), which is not a member of SO(n).
If you want to restrict M to be the same type of matrix as X,
then you
need to introduce your own method of taking expectations. To
my
knowledge (which is not necessarily that broad) there is no
standard
method for de[CapitalThorn]ning such an expectation.
-MO
===
Subject: Re: EXPECTATIONS OF A MATRIX
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EFdtl11282;
>HI!
>SORRY FOR MY IGNORANCE
>BUT HOW CAN I COMPUTE THE EXPECTATION OF A MATRIX?
>IN PARTICULAR:
>IF:
>MATRIX is a (m * n) matrix
> E(...) is the expectation operator
>in practical terms, how can I calculate E(MATRIX)?
>You simply [CapitalThorn]nd the expectation of each element. Let X =
[x_ij] be
>the random matrix in question. Then its expected value M =
E(X) is
>just the matrix of expected values of the elements of X:
>M = [ E(x_ij) ]
>Of course, you need to know the probability density function
(pdf),
>p_ij, of each element.
>The only problem with the answer above is that M may not be
the same
>kind of matrix as X. For example, suppose X is a member of
SO(n),
the
>special orthogonal matrices ( (X)(X^t) = (X^t)(X) = I and
det(X) =
>+1). M is, in general, not a member of SO(n). To make the
example
>more speci[CapitalThorn]c, suppose X is a member of SO(2), and that it is
>uniformly distributed on SO(2). Then X can be represented by
>X =
>[ cos(theta) -sin(theta) ]
>[ sin(theta) cos(theta) ]
>where theta is uniformly distributed on the interval [0,
2pi]. It
>should be easy to see that M = E(X) = 0^(2 x 2) (the 2 x 2
matrix of
>zeros), which is not a member of SO(n).
>If you want to restrict M to be the same type of matrix as
X, then
you
>need to introduce your own method of taking expectations. To
my
>knowledge (which is not necessarily that broad) there is no
standard
>method for de[CapitalThorn]ning such an expectation.
>-MO
in the basic linear regression case (homoscedastic errors)
how can I calculate the Expectation of the Matrix (n x n) of
the
residual terms, named variance-covariance matrix of the error
terms?
I could only assume that it is:
var(res) x I
where: I is the identity matrix
and var(res) is a scalar : the variance of the error term
===
Subject: Re: Another problem on Stokes's Theorem
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECcij26769;
Anyone with any suggestons about how to solve this problem?
Michelle
===
Subject: Governing Equation and Boundary Conditon for Polar
orthotropic
circular plate.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECckL26863;
Can you provide me the governing equation and boundary
condition of
polar orthotropic cicular plates..Pls. send me the
formulation.
===
Subject: Governing Equation and Boundary Conditon for Polar
orthotropic
circular plate.
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ECckG26851;
Can you provide me the governing equation and boundary
condition of
polar orthotropic cicular plates..Pls. send me the
formulation.
===
Subject: Phrase Structure
Can someone please solve the following
Construct phrase-structure grammars to generate the following
set
{0^n1^2n | n >= 0}
===
Subject: Re: Phrase Structure
alt.math.undergrad:
> Can someone please solve the following
> Construct phrase-structure grammars to generate the
following set
> {0^n1^2n | n >= 0}
(I assume that this should be {0^n 1^(2n) : n >= 0}.) This
is really extremely straightforward; you should be able to
do it yourself. Here are a few hints, though. You can do
it with a single non-terminal symbol; the main production
should generate one new 0 and two new 1's. Have you seen a
grammar that generates {0^n 1^n : n >= 0}? If so, just
modify it slightly.
Brian
===
Subject: Truth Tables Help
I have a four part question in a tutorial, parts 1 & 2 I
don't have any
problem with. Parts 3 & 4 I have no idea what they mean. Can
someone
explain
to me what parts 3 & 4 mean and how can I answer the
questions.
Alternatively if you can point me to a website that might be
of some help.
The question:
Draw the truth tables for each of the following expressions:
1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
2. (A + B' + C' + D)(A' + B + C 
+ D')
3. Sm(2.5.6.9)
4. ?M(0,7,10,11)
Appreciate any help.
TIA
===
Subject: Truth Tables Help
My earlier post should read:
I have a four part question in a tutorial, parts 1 & 2 I
don't have any
problem with. Parts 3 & 4 I have no idea what they mean. Can
someone
explain
to me what parts 3 & 4 mean and how can I answer the
questions.
Alternatively if you can point me to a website that might be
of some help.
The question:
Draw the truth tables for each of the following expressions:
1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
2. (A + B' + C' + D)(A' + B + C 
+ D')
3. Sigma m(2.5.6.9)
4. Pi M(0,7,10,11)
3 & 4 should read:
_
/_ m (2.5.6.9)
__
I I M (0,7,10,11)
Appreciate any help.
TIA
===
Subject: Re: Truth Tables Help
> My earlier post should read:
> I have a four part question in a tutorial, parts 1 & 2 I
don't have any
> problem with. Parts 3 & 4 I have no idea what they mean.
Can someone
explain
> to me what parts 3 & 4 mean and how can I answer the
questions.
> Alternatively if you can point me to a website that might
be of some
help.
> The question:
> Draw the truth tables for each of the following expressions:
> 1. AB'CD' + 
ABC'D' + A'BC'D + 
A'B'CD
> 2. (A + B' + C' + D)(A' + B + 
C + D')
> 3. Sigma m(2.5.6.9)
> 4. Pi M(0,7,10,11)
> 3 & 4 should read:
> /_ m (2.5.6.9)
> __
> I I M (0,7,10,11)
What do your texts or lecture notes say these notations mean?
Guess:
Sigma indicates + over some set, and Pi indicates . over some
set--but
what sets?
===
Subject: numeric series question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9EJJV501477;
We know that:
inf
---
n+1 1 1 1 1
(-1) ----- = 1 - --- + --- - --- + ... = eta(1) = ln2
/ n 2 3 4
---
n=1
from the expansion of ln(1+x);
we also know that:
inf
---
n+1 1 1 1 1
(-1) ----- = 1 - --- + --- - --- + ... = pi/4
/ 2n-1 3 5 7
---
n=1
from the expansion of atan(x);
but what about:
inf
---
n+1 1 1 1 1
(-1) ----- = 1 - --- + --- - --- + ... = ???
/ 3n-2 4 7 10
---
n=1
more generally,
inf
---
n+1 1
(-1) -------- = ??? [k integer >=1] (A)
/ kn-(k-1)
---
n=1
It's interesting to note that the family of alternating
series (A)
above,
is connected with
inf
---
1 1
-------------- = --- [for every integer k>=1] (B)
/ (kn-k+1)(kn+1) k
---
n=1
In order to see how, take k=1: the series (B) is: inf
---
1 1 1 1 1 1 1
----- + ----- + ----- + ----- + ----- + ----- + ... = -------
= 1
1*2 2*3 3*4 4*5 5*6 6*7 / n(n+1)
---
n=1
Now the odd terms make up the series (A) with k=1:
inf
---
1 1 1 1 n+1 1
----- + ----- + ----- + ----- + ... = (-1) --- = ln2
1*2 3*4 5*6 6*7 / n
---
n=1
because (1 - 1/2) = 1/2; (1/3 - 1/4) = 1/12 etc.
(1/n - 1/(n+1)) = 1/n*(n+1)
The even terms make up another subseries, with sum = 1-ln2
For k>1 there is a coef[CapitalThorn]cient (which is exactly 1/k) to take
out...
eg. k=2 we want to know the Ôodd' subseries of 
(B),
1/(1*3) + 1/(5*7) + 1/(7*9) + ...
well,
1 1 1
1 - --- --- - ---
1 3 1 5 7
----- = --------- , ----- = ----------- , etc.
1*3 2 5*7 2
because we know that
1-(1/3)+(1/5)-(1/7)+ etc. = pi/4,
our subseries is pi/8 !
In the end, for k=1, series (B) is split in two series, one
with sum =
ln2,
another with sum = 1-ln2.
For k=2 the similar split gives a series with sum = pi/8 and
another
with sum
(1/2 - pi/8).
What is the Ôsplit' for k>=3, k integer?
That is, which is the sum of the Ôodd' subseries 
of (B), for k
integer, k>=3?
In order to calculate this you need to know the sum of (A)
for k>=3,
but,
how much is it?
Is it known or is it still unknown in its concise form, like
zeta(2k+1)?
===
Subject: Re: numeric series question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ELLBc12432;
Here's a hint.
Use the expression for ln(1+x) where x=i, and where x=-i.
See if you can combine the series for ln(1+i) and for ln(1-i)
to get the alternating sum
1/1 - 1/3 + 1/5 - 1/7 + 1/9 -+ ... .
(Subtract the two series to get some unwanted terms to
cancel.)
Notice this is the same route that we use to get
1/1 - 1/2 + 1/3 - 1/4 + 1/5 -+ ... .
For the series you asked about,
1/1 - 1/4 + 1/7 - 1/10 + 1/13 - 1/16 +- ...
you want the coef[CapitalThorn]cients of 1/n to be periodic with period 6:
1,0,0,-1,0,0,1,0,0,-1,0,0,1,...
This will happen if you look at ln(1+x)
where x is a sixth root of unity.
You have six choices here: if w=exp(2 pi i / 6),
we can let x=w, w^2, w^3=-1, w^4, w^5, or w^6=1.
(The last one doesn't help.)
Write out those six series, and see whether you can [CapitalThorn]nd some
linear combination of them that gives the series you want.
Then generalize to the other values of k.
===
Subject: Descrete Math help with question
by support1.mathforum.org (8.11.6/8.11.6/The Math Forum,
$Revision:
1.9 primary) id i9ELLBD12436;
I'v been trying to [CapitalThorn]gure out this question and 
i don't even
know
where to begin.
De[CapitalThorn]ne two Sets A and B as follows: A = {(2n+1)^3|n E Z} and
B = {(2n+1)|n E Z}
a) prove that A is a proper subset of B.
b) suppose we redi[CapitalThorn]ne A and B, replacing Z by R . What is the
relation between these two sets? State and prove your answer.
===
Subject: Re: Descrete Math help with question
days. My association with the Department is that of an
alumnus.
>I'v been trying to [CapitalThorn]gure out this question and 
i don't even
know
>where to begin.
>De[CapitalThorn]ne two Sets A and B as follows: A = {(2n+1)^3|n E Z} and
>B = {(2n+1)|n E Z}
>a) prove that A is a proper subset of B.
The [CapitalThorn]rst set, A, contains the cubes of the odd integers. So 
it
contains 1 (=1^3), -1 (=-1^3), 27 (=3^3), -27 (=(-3)^3), 125,
-125,
etc.
Set B contains the odd integers.
So: can you show that
(a) every element of A is an element of B (i.e., A is
contained in
B)?
For example, take an arbitrary element of A, (2k+1)^3 = 8k^3 +
12k^2 + 6k + 1. Can you should that it can be written as 2n+1
for
some integer n? If so, then it would necessarily be in B.
(b) That there is at least one element of B which is not in A
(so A
is not equal to B)? For example: when n=1, you get the element
(2+1)=3 in B. Can you show that there is NO integer k such
that
(2k+1)^3 = 3?
>b) suppose we redi[CapitalThorn]ne A and B, replacing Z by R . What is 
the
>relation between these two sets? State and prove your answer.
So now A contains all real numbers which can be expressed as
(2y+1)^3,
and B contains all real numbers which can be expressed as
(2x+1), with
x and y any real numbers.
IS there any relation between A and B now? Is A contained in
B? Is A
properly contained in B? Is B contained in A? Is B properly
contained
in B?
--
It's not denial. I'm just very selective 
about
what I accept as reality.
--- Calvin (Calvin and Hobbes)
Arturo Magidin
magidin@math.berkeley.edu